The Mixing Problem: Purification and
God does not play dice with the universe.
Economic theory stresses that a proposed mechanism for solving a coordination problem assuming self-regarding agents is plausible only if it is incentive compatible: each agent should find it in his interest to behave as required by the mechanism. However, a strictly mixed-strategy Nash equilibriumfails to be incentive compatible, because a self-regarding agent i is indifferent to any mixed strategy in the support of This chapter deals with the solution to this problem. We conclude that, while ingenious justifications of the incentive compatibility of mixedstrategy Nash equilibria have been offered, they fail except in a large majority of cases. We suggest that the solution lies in recognizing both the power of social norms, and of a human other-regarding psychological predisposition to conform to social norms even when it is costly to do so.
In Throwing Fingers (§2.7), there is a unique mixed-strategy Nash equilibrium in which both players choose each of their pure strategies with probability 1/2. However, if both pure strategies have equal payoffs against the mixed strategy of the other player, Why bother randomizing? Of course, this problem is perfectly general. By the fundamental theorem (§2.5), any mixed strategy best response consists of equal-payoff pure strategies, so why should a player bother randomizing? Moreover, this argument holds for all other players as well. Therefore, no player should expect any other player to randomize. This is the mixing problem.
We assume that the game is played only once (this is called a one-shot game, even though it could be an extensive form game with many moves by each player, as in chess), so there is no past history on which to base